Some Results Of Monotone Dynamics With Applications

This paper focuses on monotone dynamics,both the quasimonotone condition for K-type systems is relaxedand the dynamics of skew-product semiflows generated by nonautonomous differential equations is studied.The contents are divided into autonomous and nonautonomous cases.In autonomous case,the quasimonotone conditions which have known are not satisfied in applications.A typical example is the Hopfield-type neural networks with the interaction among neurons being not only excitatory but also inhibitory.Motivated by this,we introduce a weak quasi-monotone condition and consider the family of sets parametrized by K-type monotone matrices.It is easy to see the sets are closed cones of state spaces. Thus,a new ordering is generated.Under the assumption that weak quasimonotone condition holds,we prove that generic solutions converge to equilibrium and there exists an ordering convex convergent point set.At last,as an application,we apply our results to a delay Hopfield-type neural networks.In nonautonomous case,we consider the dynamics of skew-product semiflows.First,under the assumptions that skew-product semiflows satisfy suitable monotonicity and concavities,the limit set trichotomy and dichotomy for skew-product semiflows is established,which gives a complete description of the long-time behavior of trajectories. As an application,a delayed Hopfield-type neural network model is considered.Since the omega limit set of the skew-product semiflow,especially its 1-cover property,plays an important role in considering its dynamics,inheriting the above assumptions of skewproduct semiflows,we consider the omega limit set of skew-product semiflows.Under the assumption that two omega limit sets satisfy completely strong ordering,we conclude that a big one is a copy of the base,a global or partial attractor,which make us having a further understanding for the structure of the skew-product semiflow.At last,we apply our results to a delay differential equations.Next,we study exponential dichotomy of linear skew-product semiflows which come from linearizing skew-product semiflows on a compact positively invariant subset M of semiflows and construct a necessary and sufficient condition of M being hyperbolically stable in the form of skew-product semiflows admitting an exponential dichotomy and the relationship between continuous separation and exponential dichotomy under assumptions that skew-product semiflows are eventually strongly monotone.At last,we list a sufficient and necessary condition for exponential dichotomy of linear skew-product semiflows in terms of the admissibility of the pair(B(R⁺,X),B₀(R⁺,X)). At last,skew-product semiflows induced by semi-convex and type-K competitive almost periodic delay differential equations are studied.If M is a compact positively invariant subset of the skew-product Semiflow,then continuous separation of the skewproduct semiflow on M holds.Furthermore,if two minimal subsets M₁ and M₂ of the skew-product semiflow satisfy eventually strongly type-K ordering M₁ <<(?)M₂,then M₁ is an attractor.Finally,these results are applied to a nonautonomous delayed Hopfield-type neural networks with a diagonal-nonnegative type-K monotone interconnection matrix and sufficient conditions are obtained for the existence of global or partial attractors.